We want to understand which building characteristics influence energy demand and quantify the expected energy consumption.

Work with smart meter and characteristic data for 99 commercial buildings across Australia.

3 years 15-minute meter data available.

Worked with engineers to identify which characteristics are suspected to be most important factors in energy consumption.

Working with strongly correlated data within buildings. Use linear mixed model with buildings treated as random effects and characteristics as fixed effects. A random slope based off year used to capture any trends in building consumption.

For a particular season and hour of the day, the demand \(y_{ij}\) for building \(j\) and observation \(i \in \left\{ 1, 2, \ldots n_j\right\}\) is

\[ \log y_{ij} = \beta_0 + \sum_{h=1}^p \beta_h x_{hij} + u_{0j} + u_{1j} t_{ij} + \epsilon_{ij}, \\ \begin{bmatrix} u_{0j} \\ u_{1j} \end{bmatrix} \sim N(0, \Omega_u), \quad \Omega_u = \begin{bmatrix} \sigma^2_{u0} & \sigma_{u01} \\ \sigma_{u01} & \sigma^2_{u1} \end{bmatrix}, \\ \epsilon_{ij} \sim N(0, \sigma^2_\epsilon), \]

where \(\beta_h\) is the coefficient for predictor \(x_{hij}\), \(u_{0j}\) is the random intercept for building \(j\), \(u_{1j}\) is the random slope coefficient for year \(t_{ij}\) and \(\epsilon_{ij}\) are the residuals.

Estimating energy profiles

Consider attribute \(h\) for building \(j\) changing from false to true. Then we can use the approximation

\[ Y_{ij} |_{X_{hij} = 1} \approx (Y_{ij} |_{X_{hij} = 0}) e^{\beta_h} \]

The values \(e^{\beta_h}\) can be interpreted as the percentage change when \(X_{hij}\) changes value from false to true.

For our model selection we use the marginal AIC (mAIC) which is the most widely used information criteria for mixed effects models. Vaida and Blanchard (2005) define the mAIC as

\[ mAIC = -2\ell\left(\hat{\mathbf{\beta}}\right) + 2 (p+q), \]

where \(\ell\left(\hat{\mathbf{\beta}}\right)\) is our log-likelihood function, \(p\) is the number of fixed effects, \(q\) is the number of random effects. We choose this criterion as it is both simple to understand and has been used in many studies (Müller, Scealy, and Welsh 2013).

Unfortunately there was no clear best model using \(mAIC\). Our problem can be approached using multimodel inference.

Candidate sets

To construct our candidate set we must first determine the probability of each candidate model. To do so we use Akaike weights. Given \(R\) candidate models the Akaike weight for model \(g_i\) is:

\[ w_i = \frac{\mathcal{L} \left(g_i | \mathbf{x} \right)}{\sum_{r=1}^R \mathcal{L} \left(g_r | \mathbf{x} \right)} = \frac{e^{-\frac{1}{2}\Delta_i}}{\sum_{r=1}^R e^{-\frac{1}{2}\Delta_r}}, \]

where \(\mathcal{L} \left(g_i | \mathbf{x} \right)\) is the likelihood of model \(g_i\) given data \(\mathbf{x}\) and \(\Delta_i = mAIC_i - mAIC_{min}\) and is referred to as the AIC difference. We use \(mAIC\) in our analysis as we are dealing with mixed effects models. Other information criteria for mixed effects models such as the conditional AIC (Greven and Kneib 2010; Vaida and Blanchard 2005) may also be used.

Parameter estimates

Parameters are estimated by "averaging" the models in our confidence set. There are two common approaches:

• Natural-model averaging averages over all candidate models where a parameter of interest occurs.
• Full-model averaging considers all candidate models. If a variable is not selected in one of the candidate models, full-model averaging sets its estimate to zero.

Simulation studies have found that full-model averaging can help to reduce problems caused by model selection bias towards over-complex models (Lukacs, Burnham, and Anderson 2010).

Given a candidate set of \(R\) models our coefficients \(\beta_h\) are estimated by full model averaging:

\[ \hat{\bar{\beta_h}} = \sum^R_{i=1} w_i \hat{\beta}_{hi}, \]

where \(\hat{\beta}_{hi}\) is the estimate of \(\beta_h\) based on model \(g_i\). If \(\beta_h\) is not chosen in model \(g_i\) then \(\hat{\beta}_{hi}\) is defined to equal zero in the above formula.

Unconditional confidence intervals

Once a set of candidate models has been identified we can construct confidence intervals that reflect both parameter and model selection uncertainty. The \(\left( 1-\alpha \right)100\%\) unconditional confidence intervals for the a model averaged coefficient \(\hat{ \bar{ \beta}}_h\) is given by

\[ \hat{ \bar{ \beta}}_h \pm z_{1- \alpha/2} \widehat{ \text{ase}}\left( \hat{ \bar{\beta}}_h \right), \]

where \(\widehat{ \text{ase}}\left( \hat{ \bar{\beta}}_h \right)\) is the adjusted standard error from Burnham and White (2002). It is given by

\[ \widehat{ \text{ase}}\left( \hat{ \bar{\beta}}_h \right) = \sum^R_{i=1} w_i \sqrt{ \left( \frac{t_{\text{df}_i, 1-\alpha/2}}{z_{1-\alpha/2}} \right)^2 \widehat{\text{var}} \left( \hat{\beta}_{hi} | g_i \right) + \left( \hat{\beta}_{hi} - \hat{\bar{\beta}}_h \right)^2 }, \]

where \(\bar{\beta}_h\) is the model averaged estimator of \(\beta_h\), \(\widehat{\text{var}} \left( \hat{\beta}_{hi} | g_i \right)\) is the estimated variance of parameter \(\beta_{hi}\) in model \(g_i\), and \(w_i\) are weights. The calculation of \(\widehat{\text{var}} \left( \hat{\beta}_{hi} | g_i \right)\) for mixed effects models is reasonably complex and is omitted (see Bates et al. (2015) for a discussion).

Including more buildings will allow for more confident conclusions to be reached and more characteristics to be investigated.

• Apply to the residential sector to estimate the impact of demand influences such as solar generation, batteries or other household items.
• Assess the impact of policies on energy consumption.